Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Dimension of compositum of field extensions

Suppose we have fields $L$, $M$ and $N$ all infinite algebraic Galois extensions of a field $k$ such that $L \cap M$, $L \cap N$ and $N \cap M$ are finite dimensional extensions of $k$. Then is $L ...
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Showing that $[\mathbb C(z):\mathbb C(f_0)]=$topological degree of $f_0\in\mathbb C(z)$ as a function from the Riemann sphere to itself

McKean & Moll offer the following sketch of a proof. Let $P(x)=a_0(x)-f_0b_0(x)$ in $\mathbb C(f_0)[x]$ where $f_0(z)=\dfrac{a_0(z)}{b_0(z)}$ is in $\mathbb C(z)$. Then evidently, $\deg P=\#$ ...
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1answer
148 views

Does product of Galois groups equal to the Galois group of corresponding fields intersection?

Let $k$ be a field, $\bar{k}/k$ be a Galois extension with $G=Gal(\bar{k}/k)$ be an Abelian group(may be infinite). If $K,L$ are intermediate fields, denote $G_K=Gal(\bar{k}/K), G_L=Gal(\bar{k}/L)$. ...
3
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1answer
340 views

Does every field have a non-trivial Galois extension?

Clearly a field $F$ with no Galois extensions must have only non-separable elements in any extension (otherwise, take the minimal polynomial of some separable element over $F$ - its splitting field ...
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1answer
61 views

Subgroups of groups of order $2^{a-1}$

The context here is the following exercise Let $m=2^a$ with $a > 2$. Show that $\mathbb{Q}(\theta_m)$ contains exactly three quadratic subfields. By Galois theory, this reduces to the problem ...
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2answers
560 views

About cyclotomic extensions of $p$-adic fields

I've been working on the problem of finding the maximal abelian extension of $\mathbb{Q}_5$ that is killed by $5$. In other words, find the abelian extension of $\mathbb{Q}_5$ with Galois group ...
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2answers
323 views

Field $\mathbb{Q}(\sqrt{3}, \sqrt{-1})$

Am I right to say that the field $\mathbb{Q}(\sqrt{3}, \sqrt{-1})$ is an algebraic extension of $\mathbb{Q}$? Because $\mathbb{Q}\subset\mathbb{Q}(\sqrt{3})\subset\mathbb{Q}(\sqrt{3})( ...
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3answers
486 views

Are there (finite) field extensions which aren't Galois?

Let $K=F(\alpha)$. It strikes me that there are two possibilities: $\alpha \in F$, in which case $K\cong F$ $\alpha\not\in F$, in which case $K$ is the splitting field of $(x-\alpha)$ (and hence ...
2
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1answer
267 views

Degree of a field extension when compared to a Galois group

Consider $K=Q(\sqrt{2},\sqrt{3})$ - I think $K$ is Galois since it's the splitting field of $(x-\sqrt 2)(x+\sqrt 2)(x-\sqrt 3)(x+\sqrt 3)$. I feel like $G(K/F)$ is isomorphic to the Klein 4 group ...
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439 views

The radical solution of a solvable 17th degree equation

(The question is at the bottom of the post.) Here's a "natural" solvable 17-th deg eqn with small coefficients: $$\begin{align*} x^{17}-6 x^{16}&-24 x^{15}-42 x^{14}-31 x^{13}-23 x^{12}-7 ...
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2answers
722 views

Galois group of $x^6+3$ over $\mathbb Q$

I'm having some difficulties finding the Galois group of the polynomial $g(x)=x^6+3$ over $\mathbb Q$. Here's what I did : I observed that the roots of the given polynomial are $\sqrt[6]3 ...
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3answers
211 views

A simple question about Galois groups

When talking about field extensions of degree two I understand the automorphisms in the Galois group intuitively as analogous to complex conjugation. I lose my understanding when going to field ...
4
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2answers
139 views

Galois Theory Problem

Let $L$ be a Galois extension of fields such that $Gal(L/K)=GL_{2}(\mathbb{F}_{p})$. Let $L_{1}$ and $L_{2}$ be subfields of $L$ containing $K$ corresponding to subgroups ...
2
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1answer
114 views

Double root mod p implies transposition in galois group?

Let $K$ be the splitting field of some irreducible polynomial $f(x)$ in $\mathbb{Z}[x]$, and let $B$ be the integral closure of $\mathbb{Z}$ in $K$. Suppose that, for some prime $p$: $f(x) \equiv ...
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1answer
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Intuition behind looking at permutations of the roots in Galois theory

What I find after reading books is that they explain only the conceptual definition and no one mentions the explanation behind it; I have been reading the Galois theory as many people told me to read, ...
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1answer
116 views

Classification of field extensions

Is it true that all extension of field $k$ are subfields of $\bar k$ (algebraic closure of $k$)? The same question for all algebraic extension. Thanks.
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162 views

Action of Galois group on set of generators

Let $k\subset K$ be a Galois extension, i.e. $$K=\langle{\xi}_{1}, {\xi}_{2},\ldots, {\xi}_{n}\rangle,$$ $$k\supset K=\{a_0+a_1{\xi}_{1}+\ldots+a_n{\xi}_{n}|a_0,\ldots,a_n\in k\}.$$ Is it true that ...
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1answer
108 views

Do we need to distinguish between positive and negative square roots?

I am seeking resolution of a disagreement over the nature of Galois theory. In the followup comments on a separate thread, I made the claim that the beauty of Galois theory is that it does not require ...
4
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1answer
245 views

Did Galois show $5^\sqrt{2}$ can't solve a high-order integer polynomial?

Suppose you have an unlimited quantity of the number one, and the operators plus, minus, multiply, divide, and power. Consider the (countable) set $S$ you generate by combining these: Using just one ...
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0answers
79 views

Is it true that $\forall G\leq Aut(\mathbb{P}^1)=PGL_2(\mathbb{C})$ the map $\mathbb{P}^1\rightarrow \mathbb{P}^1/G$ is defined over $\mathbb{Q}$?

Is it true that for every finite $G\leq Aut(\mathbb{P}^1_{\mathbb{C}})=PGL_2(\mathbb{C})$ the morphism $\mathbb{P}^1_{\mathbb{C}}\rightarrow \mathbb{P}^1_{\mathbb{C}}/G$ descends as a morphism (not ...
4
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1answer
431 views

What is a Weil-Deligne representation?

Can anyone explain (or give reference) Weil-Deligne representations, and how they are linked with Galois representations of p-adic field (on l-adic vector spaces or else) ?
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1answer
66 views

$\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})$, where $m|n$

Find $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})$, where $m|n$. Is this proof correct? Since the base field is finite, $\mathbb{F}_{p^m}\subset \mathbb{F}_{p^n}$ is a Galois extension. (Is ...
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1answer
149 views

Galois group $Gal(\mathbb{Q}[\xi]:\mathbb{Q})$

Lets $\xi$ is primitive n-th root of unity over $\mathbb{C}$ and lets $$\mathbb{Q}[\xi]\ni\alpha=\sum_{i=0}^{n-1}a_i{\xi}^i.$$ Consider any element $g\in Gal(\mathbb{Q}[\xi]:\mathbb{Q})$. ...
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5answers
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Galois group of $x^3 - 2 $ over $\mathbb Q$

I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with ...
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4answers
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Do finite algebraically closed fields exist?

Let $K$ be an algebraically closed field ($\operatorname{char}K=p$). Denote $${\mathbb F}_{p^n}=\{x\in K\mid x^{p^n}-x=0\}.$$ It's easy to prove that ${\mathbb F}_{p^n}$ consists of exactly $p^n$ ...
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4answers
405 views

How to interpret the phrase “transforms under the irreducible representation”?

I'm reading Robert Gilmore's "Lie Groups, Physics, and Geometry," and trying to understand his brief presentation of Galois theory. I think I get the gist of the method, but would be grateful for help ...
4
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1answer
393 views

Normal closure of a radical extension is radical

I'm struggling to understand the proof that the normal closure of a radical extension of fields is also a radical extension, which is crucial since it allows us to work with radical and normal ...
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2answers
994 views

Exercises on Galois Theory

I need a source for exercises on classical Galois Theory, or to be more specific, Galois extensions of finite fields and the rationals as well as applications (solvability by radicals, for example). ...
11
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1answer
568 views

Galois group of a reducible polynomial over $\mathbb {Q}$

Let $f \in\mathbb {Q}[X]$ be reducible - for the sake of simplicity, $ f = gh$ with $g,h \in\mathbb {Q}[X]$ irreducible. Let L be the splitting field of f. Does $Gal(f) \simeq Gal(g) \times Gal(h)$ ...
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6answers
3k views

How to show that $\sqrt{2}+\sqrt{3}$ is algebraic?

In Abbot's Understanding Analysis I am asked to show that $\sqrt{2}+\sqrt{3}$ is an algebraic number. I have shown that those two are algebraic separately (that was simple), but I can't figure out ...
7
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1answer
628 views

Reference book for Artin-Schreier Theory

The aim of the question is very simple, I would like to study Artin-Schreier Theory, but I have had embarassing difficulties in finding a book which could help me in doing that. In specific I'm ...
3
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1answer
188 views

Galois action on central simple algebras

Suppose we have the Galois field extensions $\mathbb{F} \leq \mathbb{L} \leq \mathbb {K}$ with Galois groups $\Gamma = Gal(\mathbb{K} / \mathbb{F})$, $N=Gal (\mathbb{K} / \mathbb{L})$. The Galois ...
2
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1answer
158 views

Galois theory: splitting field of cubic as a vector space

I'm trying to express the splitting field of a cubic equation as a vector space over the rationals. Specifically I am looking for a set of six independent vectors that span the space. If the roots are ...
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3answers
381 views

Galois ring extension

Is there an analogous theory to Galois extension of fields for commutative rings? In particular, what does it mean for a ring extension to be Galois? Thanks.
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0answers
129 views

Extensions Q(α,β) and Q(α∗β) over Q are the same one?? When?

I would like to know in which conditions the extensions $\mathbb{Q}(\alpha,\beta)$ and $\mathbb{Q}(\alpha*\beta)$ over $\mathbb{Q}$ are the same one. Thanks in advance.
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3answers
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How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
3
votes
1answer
247 views

The degree of the algebraic closure over the separable closure of an imperfect field

Let $K$ be imperfect, $K^a$ its algebraic closure and $K^{\rm sep}$ its separable closure. Show $[K^a \colon K]$ and $[K^a\colon K^{\rm sep}]$ are infinite. Is $[K^{\rm sep}\colon K]$ infinite? ...
2
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1answer
63 views

Basis of Subextension

I'm investigating the splitting field of $x^{17}-1$. Clearly this is just $\mathbb{Q}(\zeta)$ where $\zeta$ is a complex 17th root of unity. I know that there is a unique subextension $F$ of order 2, ...
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5answers
664 views

Shortest way of proving that the Galois conjugate of a character is still a character

Let $G$ be a finite group and $\chi$ a character of $G$. The values of $\chi$ generate an abelian Galois extension $K$ of $\mathbb{Q}$, and so one can consider the conjugate $\sigma(\chi)$ of $\chi$ ...
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1answer
131 views

How to extend Galois character?

Let $D_p$ be a decomposition group of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ above $p$ (for all $p$) and $I_p$ the inertia group. Let $\chi_p$ be a character of $D_p$, such that $\chi_p$ is trivial ...
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1answer
310 views

There are enough Galois extensions?

There are enough Galois extensions? For me enough means that every finite extension of a certain field is included in a Galois extension of that field, formally: "Given a field $k$, and a finite ...
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1answer
121 views

Does there exist an English translation of Distler's paper on solving polynomials by radicals?

The Radiroot package for GAP by Andreas Distler solves by radicals polynomials with solvable Galois group. The package uses the algorithm described in Distler's paper "Ein Algorithmus zum Lösen einer ...
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2answers
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Prove that the Galois group of $x^5+x^4-4x^3-3x^2+3x+1$ is cyclic of order $5$

I have been asked to prove that the Galois group of $x^5+x^4-4x^3-3x^2+3x+1$ (presumably over $\mathbb{Q}$ is cyclic of order $5$. Unfortunately, I have no idea where to start. The hint says to show ...
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617 views

A question concerning polynomial solvable by radicals

We know from Galois Theory that a polynomial is solvable by radicals if and only if its Galois group is solvable. On the other hand solvable by radicals for example means that the equation $X^n-1=0$ ...
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2answers
1k views

Calculating the Galois group of an (irreducible) quintic

On my homework I have been asked to compute the Galois group of a quintic. I have no idea how to do this, except (a) I calculated that it was irreducible (brute-force) (b) Since it is irreducible, ...
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3answers
352 views

Complex Galois Representations are Finite

In A First Course in Modular Forms, Diamond and Shurman leave as an exercise ($9.3.3$) that every complex Galois representation is finite. While I think I have worked through this exercise here, this ...
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2answers
112 views

Geometric distinction between real cubics with different Galois group?

The following Cubics have 3 real roots but the first has Galois group $C_3$ and the second $S_3$ $x^3 - 3x + 1$ (red) $x^3 - 4x + 2$ (green) Is there any geometric way to distinguish between the ...
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2answers
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Does a cubic polynomial with 3 real roots have Galois group C3?

If an irreducible cubic polynomial with coefficients in $\mathbb Q$ has Galois group $C_3$ then, since no order $2$ symmetry lies in the Galois group no complex conjugation acts on the roots, it's ...
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3answers
1k views

Degree 2 Field extensions

Are all degree 2 field extensions Galois? I know that this is true over the rationals. But is it true in General?
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2answers
1k views

“Standard” ways of telling if an irreducible quartic polynomial has Galois group C_4?

The following facts are standard: an irreducible quartic polynomial $p(x)$ can only have Galois groups $S_4, A_4, D_4, V_4, C_4$. Over a field of characteristic not equal to $2$, depending on whether ...